Title: Comparison of SMC and OMC results in determining the ground-state and meta-stable states solutions for UO$_2$ in DFT+U method

Abstract: Correct prediction of the behavior of UO2 crystal, which is an antiferromagnetic system with strongly-correlated electrons, is possible by using a modified density functional theory, the DFT+U method. In the context of DFT+U, the energy of crystal turns out to be a function with several local minima, the so-called meta-stable states, and the lowest energy state amongst them is identified as the ground state. OMC was a method that were used in DFT+U to determine the ground state. Recently the SMC method was proposed which using only the oxygen electronic spin-polarization degrees of freedom also revealed the multi-minima structure of energy in the DFT+U approach and led to results in good agreement with experiment. In this work, we compare the SMC and OMC results and show that although the ground states of the two methods have similar energies and geometries, the electronic structures have significant differences. Moreover, we show that the GS obtained from SMC is by 0.0022 Ry/(formula unit) above that of OMC. The different GS results from the two methods implies that they search for the minimum-energy state over different subspaces of electron densities and each method alone is not capable to locate the global minimum of the energy. Therefore, to obtain the global-minimum state of energy one has to search over larger subspaces that involve both occupation matrices of U atoms and starting magnetization of O atoms.